A Constructive Analysis of Convex-Valued Demand Correspondence for Weakly Uniformly Rotund and Monotonic Preference

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Volume 2011, Article ID 960819,7pages doi:10.1155/2011/960819

Research Article

A Constructive Analysis of Convex-Valued Demand Correspondence for Weakly Uniformly Rotund and Monotonic Preference

Yasuhito Tanaka

¹

and Atsuhiro Satoh

²

1Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan

2Graduate School of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan

Correspondence should be addressed to Yasuhito Tanaka,[email protected] Received 23 December 2010; Accepted 14 February 2011

Academic Editor: Chenghu Ma

Copyrightq2011 Y. Tanaka and A. Satoh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Bridges 1992 has constructively shown the existence of continuous demand function for consumers with continuous, uniformly rotund preference relations. We extend this result to the case of multivalued demand correspondence. We consider a weakly uniformly rotund and monotonic preference relation and will show the existence of convex-valued demand correspondence with closed graph for consumers with continuous, weakly uniformly rotund and monotonic preference relations. We follow the Bishop style constructive mathematics according to Bishop and Bridges1985, Bridges and Richman1987, and Bridges and Vˆıt¸˘a2006.

1. Introduction

Bridges 1 has constructively shown the existence of continuous demand function for consumers with continuous, uniformly rotund preference relations. We extend this result to the case of multivalued demand correspondence. We consider a weakly uniformly rotund and monotonic preference relation and will show the existence of convex-valued demand correspondence with closed graph for consumers with continuous, weakly uniformly rotund and monotonic preference relations.

In the next section, we summarize some preliminary results most of which were proved in1. InSection 3, we will show the main result.

We follow the Bishop style constructive mathematics according to2–4.

2. Preliminary Results

Consider a consumer who consumesNgoods.Nis a finite natural number larger than 1. Let X ⊂R^Nbe his consumption set. It is a compacttotally bounded and completeand convex

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set. LetΔbe ann−1-dimensional simplex andp∈Δa normalized price vector of the goods.

Letp_i be the price of theith good, then_N

i1p_i 1 andp_i ≥ 0 for eachi. For a givenp, the budget set of the consumer is

β p, w

≡

x∈X:p·x≤w

, 2.1

wherew > 0 is the initial endowment. A preference relation of the consumeris a binary relation onX. Letx, y ∈ X. If he prefersxtoy, we denotex y. A preference-indiﬀerence relationis defined as follows;

xy, iﬀ¬ yx

, 2.2

wherex yentailsx y, the relationsandare transitive, and if eitherx y zor xyz, thenxz. Also we have

xy iﬀ∀z∈X

yz⇒xz

. 2.3

A preference relationis continuous if it is open as a subset ofX×X, andis a closed subset ofX×X.

A preference relationonXis uniformly rotund if for eachεthere exists aδεwith the following property.

Definition 2.1uniformly rotund preference. Letε >0,x, andypoints ofXsuch that|x−y| ≥ ε, andza point ofR^Nsuch that|z| ≤δε, then either1/2x y zxor1/2x y zy.

Strict convexity of preference is defined as follows.

Definition 2.2strict convexity of preference. If x, y ∈ X,x /y, and 0 < t < 1, then either tx 1−tyxortx 1−tyy.

Bridges5has shown that if a preference relation is uniformly rotund, then it is strictly convex.

On the other hand, convexity of preference is defined as follows.

Definition 2.3convexity of preference. Ifx, y ∈X,x /y, and 0< t <1, then eithertx 1− tyxortx 1−tyy.

We define the following weaker version of uniform rotundity.

Definition 2.4weakly uniformly rotund preference. Letε >0,xandypoints ofXsuch that

|x−y| ≥ε. Letzbe a point ofR^Nsuch that|z| ≤δforδ >0 andz0every component ofz is positive, then1/2x y zxor1/2x y zy.

We assume also that consumers’ preferences are monotonic in the sense that ifx > x it means that each component ofxis larger than or equal to the corresponding component ofx, and at least one component ofxis larger than the corresponding component ofx, then xx.

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Now, we show the following lemmas.

Lemma 2.5. Ifx, y ∈X,x /y, then weak uniform rotundity of preferences implies that1/2x yxor1/2x yy.

Proof. Consider a decreasing sequenceδmofδinDefinition 2.4. Then, either1/2x y z_m xor1/2x y z_m yforz_msuch that|z_m|< δ_mandz_m 0 for eachm. Assume thatδmconverges to zero. Then,1/2x y zmconverges to1/2x y. Continuity of the preferenceclosedness ofimplies that1/2x yxor1/2x yy.

Lemma 2.6. If a consumer’s preference is weakly uniformly rotund, then it is convex.

This is a modified version of Proposition 2.2 in5.

Proof. 1Let xandy be points inX such that|x−y| ≥ ε. Consider a point1/2x y.

Then,|x−1/2x y| ≥ε/2 and|1/2x y−y| ≥ε/2. Thus, usingLemma 2.5, we can show 1/43x y xor1/43x y y, and1/4x 3y xor1/4x 3y y. Inductively, we can show that fork 1,2, . . . ,2ⁿ −1, k/2ⁿx 2ⁿ−k/2ⁿy x or k/2ⁿx 2ⁿ−k/2ⁿyy, for each natural numbern.

2Letztx 1−tywith a real numbertsuch that 0< t <1. We can select a natural numberkso thatk/2ⁿ≤t≤k 1/2ⁿfor each natural numbern.k 1/2ⁿ−k/2ⁿ 1/2ⁿ is a sequence. Since, for natural numbersmandnsuch thatm > n,l/2^m≤t≤l 1/2^mand k/2ⁿ≤t≤k 1/2ⁿwith some natural numberl, we have

l 1 2^m − l

2^m − k 1

2ⁿ − k 2ⁿ

2ⁿ−2^m 2^m2ⁿ

< 1

2ⁿ, 2.4

k 1/2ⁿ−k/2ⁿis a Cauchy sequence, and converges to zero. Then,k 1/2ⁿandk/2ⁿ converge tot. Closedness ofimplies that eitherzxorzy. Therefore, the preference is convex.

Lemma 2.7. Letxandybe points inXsuch thatxy. Then, if a consumer’s preference is weakly uniformly rotund and monotonic,tx 1−tyyfor 0< t <1.

Proof. By continuity of the preferenceopenness of, there exists a pointx x−λ such that λ 0 andx y. Then, since weak uniform rotundity implies convexity, we have tx 1−tyyortx 1−tyx. Iftx 1−tyx, then by transitivitytx 1−ty tx 1−ty−tλxy. Monotonicity of the preference impliestx 1−tyy. Assume tx 1−tyy. Then, again monotonicity of the preference impliestx 1−tyy.

LetSbe a subset ofΔ×Rsuch that for eachp, w∈S, 1p∈Δ,

2βp, wis nonempty,

3There existsξ∈Xsuch thatξxfor allx∈βp, w.

In1, the following lemmas were proved.

Lemma 2.81, Lemma 2.1. Ifp ∈Δ⊂ R^N,w ∈R, andβp, wis nonempty, thenβp, wis compact.

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Lemma 2.8with Proposition 4.4 in Chapter 4 of2or Proposition 2.2.9 of4implies that for eachp, w∈Sβp, wis located in the sense that the distance

ρ x, β

p, w

≡infx−y:y∈β p, w

2.5

exists for eachx∈R^N.

Lemma 2.91, Lemma 2.2. Ifp, w∈Sandξ βp, w(it meansξx, for allx∈βp, w), thenρξ, βp, w>0 andp·ξ > w.

Lemma 2.101, Lemma 2.3. Letp, c∈S,ξ∈Xandξβp, c. LetHbe the hyperplane with equationp·x c. Then, for eachx ∈βp, c, there exists a unique pointϕxinH∩x, ξ. The functionϕso defined mapsβp, contoH∩βp, cand is uniformly continuous onβp, c.

Lemma 2.111, Lemma 2.4. Letp, w∈S,r > 0,ξ ∈X, andξ βp, w. Then, there exists ζ∈Xsuch thatρζ, βp, w< randζβp, w.

Proof. See the appendix.

And the following lemma.

Lemma 2.121, Lemma 2.8. LetR,c, andtbe positive numbers. Then, there existsr >0 with the following property: ifp,pare elements ofR^N such that|p| ≥cand|p−p| < r,w,ware real numbers such that|w−w|< r, andyis an element ofR^Nsuch that|y| ≤Randp·yw, then there existsζ∈R^Nsuch thatp·ζwand|y−ζ|< t.

It was proved by settingr ct/R 1.

3. Convex-Valued Demand Correspondence with Closed Graph

With the preliminary results in the previous section, we show the following our main result.

Theorem 3.1. Letbe a weakly uniformly rotund preference relation on a compact and convex subset XofR^N,Δa compact and convex set of normalized price vectors (ann−1-dimensional simplex), and Sa subset ofΔ×Rsuch that for eachp, w∈S

1p∈Δ,

2βp, wis nonempty,

3There existsξ∈Xsuch thatξxfor allx∈βp, w.

Then, for eachp, w∈S, there exists a subsetFp, wofβp, wsuch thatFp, wx(it means yxfor ally∈Fp, w) for allx∈βp, w,p·Fp, w w(p·ywfor ally∈Fp, w), and the multivalued correspondenceFp, wis convex-valued and has a closed graph.

A graph of a correspondenceFp, wis

GF

p,w∈S

p, w

×F p, w

. 3.1

IfGFis a closed set, we say thatFhas a closed graph.

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Proof. 1Letp, w∈S, and chooseξ ∈Xsuch thatξ βp, w. ByLemma 2.11, construct a sequenceζ_minXsuch thatζ_mβp, wandρζ_m, βp, w<r/2^m−1withr >0 for each natural numberm. By convexity and transitivity of the preferencetζm 1−tζm 1 βp, w for 0 < t < 1 and eachm. Thus, we can construct a sequenceζ_n such that|ζ_n−ζ_{n 1}| <

εⁿ,ρζn, βp, w < δⁿ andζn βp, w for some 0 < ε < 1 and 0 < δ < 1, and so ζn is a Cauchy sequence in X. It converges to a limitζ^∗ ∈ X. By continuity of the preference closedness ofζ^∗βp, w, andρζ^∗, βp, w 0. Sinceβp, wis closed,ζ^∗∈βp, w. By Lemma 2.9,p·ζn> wfor alln. Thus, we havep·ζ^∗w. Convexity of the preference implies thatζ^∗ may not be unique, that is, there may be multiple elementsζ of βp, w such that p·ζwandζβp, w. Therefore,Fp, wis a set and we get a demand correspondence.

Let ζ ∈ Fp, w and ζ ∈ Fp, w. Then, ζ βp, w, ζ βp, w, and convexity of the preference impliestζ 1−tζβp, w. Thus,Fp, wis convex.

2 Next, we prove that the demand correspondence has a closed graph. Consider p, wandp, wsuch that|p−p|< rand|w−w|< rwithr >0. LetFp, wandFp, w be demand sets. Lety ∈ Fp, w,c ρ0,Δ> 0, andR > 0 such thatX ⊂ B0, R. Given ε >0,tδ >0 such thatδ < ε, and chooseras inLemma 2.12. By that lemma, we can choose ζ ∈ R^N such that p·ζ w and|y−ζ| < δ. Similarly, we can chooseζy ∈ R^N such that p·ζy wand|y−ζy|< δfor eachy∈Fp, w.y∈Fp, wmeansyζy. Either

|y−y|> ε/2 for ally∈Fp, wor|y−y|< εfor somey∈Fp, w. Assume that|y−y|> ε/2 for ally∈Fp, wandyζ. Ifδis suﬃciently small,|y−y|> ε/2 means|y−ζ|> ε/kand

|y−ζy|> ε/kfor some finite natural numberk. Then, by weak uniform rotundity, there existznandz_nsuch that|zn|< τn,|z_n|< τnwithτn>0,zn0 andz_n 0,1/2y ζ zn ζ and1/2y ζy z_nζyforn1,2, . . .. Again ifδis suﬃciently small,|y−ζy|< δ and|y−ζ| < δimply1/2y ζ zn yand1/2y ζy z_n y. And it follows that|1/2y ζ−1/2y ζy| < δ. By continuity of the preferenceopenness of 1/2y ζ zn y. Lety1 1/2y ζ. Consider a sequenceτnconverging to zero.

By continuity of the preferenceclosedness ofy1 yandy1 y. Note thatp·y1 w.

Thus,y1 ∈ βp, w. Sincey ∈ Fp, w, we havey1 ∈ Fp, w. Replacingy withy1, we can show thaty 3ζ/4∈Fp, w. Inductively, we obtainy 2^m−1ζ/2^m∈Fp, wfor each natural numberm. Then, we have|y−ζ|< ηfor somey∈Fp, wfor anyη >0. It contradicts

|y−ζ|> ε/k. Therfore, we have|y−y|< εorζyit means|y−ζ|< δ εandζ∈Fp, w, and soFp, whas a closed graph.

Appendix

A. Proof of Lemma 2.11

This proof is almost identical to the proof of Lemma 2.4 in Bridges1. They are diﬀerent in a few points.

Let H be the hyperplane with equationp·x w and ξ the projection of ξ onH.

Assume|ξ−ξ|> 3r. ChooseRsuch thatH∩βp, wis contained in the closed ballBξ, R aroundξand let

c

1

R

|ξ−ξ|

2. A.1

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ξ^ ϕ(x) ϕ^(x) ξ x

H H^

Figure 1: Calculation of|ϕx−ϕx|.

LetHbe the hyperplane parallel toH, betweenHandξand a distancer/2cfromH, and H the hyperplane parallel to H, betweenH and ξ and a distancer/c from H. For each x∈βp, wletϕxbe the unique element ofH∩x, ξ,ϕxthe unique element ofH∩x, ξ, andϕxthe unique element ofH∩x, ξ. Sinceξ βp, w, we haveϕxϕxxby convexity and continuity of the preference.ϕxis uniformly continuous, so

T ≡

ϕx:x∈β p, w

A.2 is totally bounded byLemma 2.8and Proposition 4.2 in Chapter 4 of2.

Sinceϕx ϕxand ϕx 1/2ϕx 1/2ϕx, we haveϕx x, and so continuity of the preferenceopenness ofmeans that there existsδ >0 such thatϕxix when|ϕx_i−ϕx| < δ. Letx1, . . . , x_nbe points ofβp, wsuch thatϕx1, . . . , ϕx_n is aδ-approximation toT. Givenxinβp, w, chooseisuch that|ϕxi−ϕx| < δ. Then, ϕxix.

Now, from our choice ofc, we have |ϕx−ϕx| < r/2 for each x ∈ βp, w. It is proved as follows. Since by the assumption|ϕx−ξ|< R,|ϕx−ξ|<

R² |ξ−ξ|². Thus, we have

ϕx−ϕx< r 2c×

R² |ξ−ξ|²

|ξ−ξ| r 2c

1

R

|ξ−ξ|

2 r

2. A.3

SeeFigure 1.

Let

t11− r 2nϕx1−ξ, η1t1ϕx1 1−t1ξ.

A.4

Then,|η1−ϕx1| r/2n,ρη1, βp, w < rn 1/2nbecause|ϕx1−ϕx1|< r/2 and ϕx1∈βp, w, and by convexity of the preferenceη1ξorη1 ϕx1.

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In the first case, we complete the proof by takingζ η1. In the second, assume that, for somek1≤k≤n−1, we have constructedη1, . . . , η_kinXsuch that

ηkϕxi 1≤i≤k, ρ

η_k, β p, w

< rn k

2n . A.5

As|ξ−ηk| > rbecause|ξ−ξ| > 3r, we can choosey ∈ ηk, ξsuch that|y−ηk| r/2n.

Then,ρy, βp, w< rn k 1/2nand eitheryξoryη_k. In the former case, the proof is completed by takingζ y. Ify ηk,y λ/2 ηk−λ/2 for allλsuch thatλ 0. Then, eithery λ/2 ϕxk 1for allλand soy ϕxk 1, in which case we setηk 1 y; or else ϕx_{k 1}η_k−λ/2 for allλand soϕx_{k 1}η_k, then we setη_{k 1}ϕx_{k 1}.

If this process proceeds as far as the construction ofηn, then, settingζηn, we see that ρζ, βp, w< rand thatζϕxifor eachi; soζxfor eachx∈βp·w.

Acknowledgment

This research was partially supported by the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research C, no. 20530165, and the Special Costs for Graduate Schools of the Special Expenses for Hitech Promotion by the Ministry of Education, Science, Sports and Culture of Japan in 2010.

References

1 D. S. Bridges, “The construction of a continuous demand function for uniformly rotund preferences,”

Journal of Mathematical Economics, vol. 21, no. 3, pp. 217–227, 1992.

2 E. Bishop and D. Bridges, Constructive Analysis, vol. 279 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1985.

3 D. Bridges and F. Richman, Varieties of Constructive Mathematics, vol. 97 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1987.

4 D. Bridges and L. Vˆıt¸˘a, Techniques of Constructive Mathematics, Springer, 2006.

5 D. S. Bridges, “Constructive notions of strict convexity,” Mathematical Logic Quarterly, vol. 39, no. 3, pp.

295–300, 1993.